Problem 67 - Calculus Early Transcendentals: Pearson New International Edition

Question

The parametric curve given by \(x=\cos a t\) and \(y=\sin b t\) is known as a Lissajous figure. The \(x\)-coordinate oscillates \(a\) times between 1 and \(-1\) as \(t\) goes from 0 to \(2 \pi\), while the \(y\)-coordinate oscillates \(b\) times over the same \(t\) interval. This behavior is repeated over every interval of length \(2 \pi\). The entire motion takes place in a unit square. Plot the following Lissajous figures for a range of \(t\) that ensures that the resulting figure is a closed curve. In each case, count the number of times that the curve touches the horizontal and vertical borders of the unit square. (a) \(x=\sin t, y=\cos t\) (b) \(x=\sin 3 t, y=\cos 5 t\) (c) \(x=\cos 5 t, y=\sin 15 t\) (d) \(x=\sin 2 t, y=\cos 9 t\)

Step-by-Step Solution

Answer

(a) 4 times; (b) horizontal 6 and vertical 8; (c) horizontal 10 and vertical 20; (d) horizontal 7 and vertical 11.

Step 1: Understand the Parametric Equations

The Lissajous figure is described by the equations \(x = \cos(at)\) and \(y = \sin(bt)\). The variables \(a\) and \(b\) dictate how many oscillations occur for the \(x\) and \(y\) coordinates, respectively, from 0 to \(2\pi\). A complete closed curve is obtained when the ratio \(a:b\) is rational.

Step 2: Determine the Range of t

To ensure that the Lissajous figure is a closed curve, \(t\) should vary from 0 to the least common multiple (LCM) of \(2\pi/a\) and \(2\pi/b\). For the given equations, \(t\) typically ranges from 0 to \(2\pi\) since \(a\) and \(b\) are integers that define complete oscillations within this interval.

Step 3: Plot the Lissajous Figure for (a)

In part (a), \(x = \sin t\) and \(y = \cos t\). This can be rewritten as \(x = \cos \left(\frac{\pi}{2} - t\right)\), making it equivalent to \(x = \cos t\) and \(y = \sin t\), a phase-shifted unit circle. The curve touches the borders four times, once on each side of the square.

Step 4: Plot the Lissajous Figure for (b)

In part (b), \(x = \sin 3t\) and \(y = \cos 5t\). The Lissajous figure completes and repeats every \(LCM(\frac{2\pi}{3}, \frac{2\pi}{5}) = 2\pi\) period. The figure will touch the horizontal banners 6 times (since \(b-a = 5-3=2\)) and the vertical borders 8 times (since \(b+a=5+3=8\)).

Step 5: Plot the Lissajous Figure for (c)

In part (c), \(x = \cos 5t\) and \(y = \sin 15t\). The Lissajous curve completes with a least common period of \(2\pi\). It touches the vertical borders 20 times (since \(b+a = 15+5 = 20\)) and the horizontal borders 10 times (since \(b-a = 15-5 = 10\)).

Step 6: Plot the Lissajous Figure for (d)

In part (d), \(x = \sin 2t\) and \(y = \cos 9t\). The least common period is \(2\pi\). The curve touches horizontal banners 7 times (since \(b-a = 9-2 = 7\)) and the vertical borders 11 times (since \(b+a = 9+2 = 11\)).

Step 7: Summarize the Border Touches

For each figure: (a) touches borders 4 times, (b) touches horizontal 6 times and vertical 8 times, (c) touches horizontal 10 times and vertical 20 times, (d) touches horizontal 7 times and vertical 11 times.

Key Concepts

Parametric Equations

Lissajous figures are fascinating patterns generated using parametric equations. In mathematics, parametric equations establish a relationship between a set of coordinates and one or more independent parameters, typically represented as variables. For Lissajous figures, the parametric equations that describe the curve are:

• \(x = \cos(at)\)
• \(y = \sin(bt)\)

Here, \( t \) is the parameter, while \( a \) and \( b \) are constants that indicate the oscillation frequencies of the \(x\) and \(y\) functions. These equations effectively map the values of \( t \) to points on a plane, producing different Lissajous figures for various \( a \) and \( b \).
By adjusting \(a\) and \(b\), you can alter the complexity and shape of the figures. If the ratio of \(a:b\) is rational (i.e., both \(a\) and \(b\) can be expressed as simple fractions), the resulting curve will be closed.

Oscillation

Oscillation, in the context of Lissajous figures, refers to the repeated up and down or back and forth movement of points along the curve. The parameters \(a\) and \(b\) in the parametric equations determine the number of times the oscillation occurs for each of the coordinates:

• The \(x\)-coordinate oscillates \(a\) times between 1 and -1.
• The \(y\)-coordinate oscillates \(b\) times.

As \(t\) progresses from 0 to \(2\pi\), these oscillations produce the intricate loops and intersections that are characteristic of Lissajous figures. Each oscillation signifies a complete transition from the maximum to minimum value or vice versa. This rhythmic phenomenon contributes to the visually appealing symmetry and elegance of the patterns, capturing the inherent motion in the geometry.

Closed Curves

A closed curve is a continuous loop with its starting and ending points coinciding, enclosing an area without any gaps. In Lissajous figures, the achievement of a closed curve depends heavily on the relationship between the frequencies \(a\) and \(b\).
For the Lissajous figures to form a closed curve, the least statistical condition is that the ratio of the frequencies \(a:b\) must be rational. This means there must be a common ratio that allows the curves to eventually reach their starting point, thus completing a loop. Without this rational ratio, the resulting figure would continue indefinitely without closure.
The process of calculating the exact range of \(t\) needed for closure involves finding a period where the components of the curve - each described by the trigonometric functions - successfully synchronize to return to the beginning. Usually, this is understood by computing the least common multiple (LCM) of their periods.

Least Common Multiple

The concept of the least common multiple (LCM) plays a critical role in ensuring that parametric equations describing Lissajous figures result in closed curves. The LCM identifies the smallest interval of \(t\) over which both oscillating functions, \(x = \cos(at)\) and \(y = \sin(bt)\), complete an integer number of cycles and thereby align to return to their initial points.
To determine the appropriate range of \(t\) for closure, calculate the LCM of the periods of the oscillations defined by \(2\pi/a\) and \(2\pi/b\). This computes the smallest value of \(t\) that satisfies both oscillation cycles simultaneously. For example, in cases where \(t\) extends from 0 to \(2\pi\), as in the exercise scenarios, the LCM helps to ascertain that the figure indeed closes after completing the requisite number of oscillations.
This mathematical analysis ensures that the Lissajous figure appears complete and coherent, resulting in the striking symmetrical patterns known to capture observers' imaginations.